Optimal. Leaf size=182 \[ \frac{2 a \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
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Rubi [A] time = 0.172281, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3878, 3872, 2838, 2564, 321, 329, 212, 206, 203, 2635, 2641} \[ -\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3878
Rule 3872
Rule 2838
Rule 2564
Rule 321
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx &=\frac{\int (a+a \sec (c+d x)) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{\int (-a-a \cos (c+d x)) \sec (c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=\frac{a \int \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \int \sec (c+d x) \sin ^{\frac{3}{2}}(c+d x) \, dx}{e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \operatorname{Subst}\left (\int \frac{x^{3/2}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ &=-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{a \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 10.7266, size = 135, normalized size = 0.74 \[ -\frac{a \left (\frac{4 \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )}{\sqrt{\sin (c+d x)}}+4 \cos (c+d x)+3 \sqrt{\csc (c+d x)} \log \left (1-\sqrt{\csc (c+d x)}\right )-3 \sqrt{\csc (c+d x)} \log \left (\sqrt{\csc (c+d x)}+1\right )+6 \sqrt{\csc (c+d x)} \tan ^{-1}\left (\sqrt{\csc (c+d x)}\right )+12\right )}{6 d e \sqrt{e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.233, size = 710, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \csc \left (d x + c\right )}{\left (a \sec \left (d x + c\right ) + a\right )}}{e^{2} \csc \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{1}{\left (e \csc{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\sec{\left (c + d x \right )}}{\left (e \csc{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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